In the last lecture. We talked about the
truth functional connective, conjunction.
We gave the truth table for conjunction.
And we showed how we could use the truth
table for conjunction to figure out which
inferences that use conjunction are valid
and which inferences are not. Today, we're
going to talk about the truth functional
connective, disjunction. We're going to
give the truth table for dis-junction, and
we're going to show how we can use that
truth table to figure out which inferences
that use dis-junction are valid and which
are not. Now in English, we usually
express disjunction by using the word or
but the word or can be used in a couple
different ways in English. For instance,
suppose that Manchester is playing
Barcelona tonight and you ask me, who's
going to win? And I say, well, I have no
idea who's going to win but I can tell you
this, it's going to be Manchester or
Barcelona. Now, what I'm suggesting when I
say, it's going to be Manchester or
Barcelona, is that it's not going to be
both. Manchester might win, Barcelona
might win. But there's no possible way
that both of them are going to win.
Sometimes, in English, when you want to
say that it's going to be one thing or the
other, but not both, you say, either or
either Manchester is going to win, or
Barcelona is going to win. But sometimes
when we use the word or, we mean it could
be one, or the other, or both. So for
instance, suppose you ask me what we
should have for dinner tonight and I say
well we could have chicken or fish. Well
there's no suggestion that we couldn't
have both maybe we could have a little bit
of chicken and a little of fish. So it has
to be chicken or fish or both. When I say
chicken or fish, I'm not suggesting it
can't be both. Sometimes in English we use
the phrase and, or to express that it
could be one or the other or both. I'll
say, we could have the chicken and, or the
fish. The truth functional connective
disjunction is expressed by the second
meaning of or. It's expressed by the
English phrase and, or where you me an it
could be one or the other or both. That's
what we're going to call disjunction in
this class. Now let's look at the truth
table for disjunction. So lets look at the
truth table for dis-junction. Suppose
you're using disjunction to combine the
propositions We eat chicken and we eat
fish into the disjunctive proposition We
eat chicken or fish. Well when is that
disjunctive proposition going to be true?
If it's true that we eat chicken, and it's
true that we eat fish, then it's going to
be true that we eat chicken or fish cuz
remember, when we use or here, we don't
mean either or, but not both. We mean and
or. Could be one, could be the other, or
could be both. So if it's true that we eat
chicken and it's true that we eat fish,
it's going to be true that we eat chicken
or fish. Now supposed it's true that we
eat chicken, but its false that we eat
fish. Well. Then, it's still going to be
true that we eat chicken or fish. Suppose
it's false that we eat chicken, but true
that we eat fish. Then, it's still going
to be true that we eat chicken or fish.
But suppose it's false that we eat chicken
and it's also false that we eat fish.
Then, is it going to be true that we eat
chicken or fish? No! Because we won't be
eating either. So then it'll be false that
we eat chicken or fish. This is the truth
table for disjunction. And, like the truth
table that we saw for conjunction, it's
going to work no matter what propositions
we put into here, or here, or here. So, no
matter what proposition you have right
here, call it P1. And, no matter what
proposition you have right here, call it
P2. When you use the truth functional
connective disjunction. To create a new
proposition out of those two
proposition's, so you got a new
proposition P one or P two. That new
disjunctive proposition is going to be
true. Whenever P1 is true, and it's also
going to be true whenever P2 is true. So
unlike conjunction. Where you need both of
the two ingredient propositions to be true
in order for the conjunctive proposition
to be true. In disjunc tion, you only need
for one of the of the two ingredient
propositions to be true in order for the
disjunctive proposition to be true. The
disjunctive proposition is false only
when. Both of the two ingredient
propositions are false. That's the only
time a disjunction is false. So now, let
me give you an example, of how you can use
the truth table for disjunction. Just show
that a particular kind of argument is
valid. We're going to discuss, a kind of
argument that is sometimes known. As
process of elimination. Here's how it
goes. Suppose, that you have to solve. A
murder mystery. Mister Jones, has been
stabbed in his living room. With a knife
in the back. Now, you figured out that
there were only two people in the house at
the time of his stabbing, the butler and
the accountant. You also know that the
knife is positioned in Mr. Johnson's back
in such a way that he couldn't possibly
have stabbed himself. So it had to be
someone else. And whoever else it was it
had to be someone who's in the house at
the time of the stabbing. So it could only
have been, the butler or the accountant,
or maybe both. So you know that the butler
did it, or the accountant did it. Now you
find out that the accountant is a
quadriplegic, so the accountant couldn't
have stabbed Mr. Jones in the back. So now
you know that the account didn't do it.
And so, from the two premises, the butler
did it, or the accountant did it. And the
accountant didn't do it. You can conclude,
the butler did it. Now, why is that
argument valid? Here's why. Think about
the truth table for disjunction again. So
remember the first premise, the butler did
it or the accountant did it is a
disjunction. It's going to be true
whenever one of it's disjuncts is true,
one of it's ingredient propositions is
true. So it's going to be true whenever
the butler did it, and it's going to be
true whenever the butler did it. The
second premise tells you that the
accountant didn't do it. So the only way
for the first premise to be true, given
that the accountant didn't do it, is fo r
the butler to have done it. And so you
know, since the accountant couldn't have
done it. That the only way for the
dis-junction, the butler did it or the
accountant did it to be true, is for the
butler to have done it and that's why you
can conclude the butler did it and your
argument is valid. That's one example of a
process of elimination argument. Of course
there are lots of others, but with all of
those others you can see why they are
valid by looking at the truth table for
dis-junction. Remember how you can use the
truth functional connective conjunction to
build a new proposition out of not just
two other propositions but sometimes three
other propositions. You can conjoin one
proposition with a second and with a
third. Well, you can do the same thing
with disjunction. You can disjoin one
proposition with a second and a third, to
create the proposition. Either this, or
that or the other or any combination of
the three. What does the truth table for
that look like? Here it is. The
disjunctive proposition, P1 or P2 or P3,
is going to be true. Whenever P1 is true,
it's also going to be true whenever P2 is
true. And it's also going to be true
whenever P3 is true. In fact, the only
time that P1 or P2 or P3, the only time
that, that disjunctive proposition is
going to be false is when all these
ingredient propositions are false. So
here's what the truth table for P1, or P2,
or P3 looks. Now let's use the truth table
for our triple disjunction to show how a
particular process of elimination argument
can be valid. Let's go back to our murder
mystery in order to do that. Now suppose
that you find out contrary to what you had
previously believed, that Butler and the
accountant were not the only people in the
house, at the time of Mr. Jonathan's
death. In addition, the maid was in the
house and the cook was in the house.
Alright. Well, now, you know, that the
butler or the maid or the cook did it. We
don't yet know which of them did it, but
we know that the butler or the maid or the
cook did it. Now suppose that yo u find
out that the maid and the cook, at the
time of the stabbing we're off in the
opposite corner of the house doing
something else together. Well now you
know, that the maid didn't do it. And you
know that the cook didn't do it. So what
can you conclude from those three
premises? Premise one, the butler or the
maid or the cook did it. Premise two, the
maid didn't do it. And premise three: the
cook didn't do it. Well, lets use the
truth table to figure this out. Premise
one of the truth table tells you that the
butler or the maid or the cook did it. So
the situation in which it falls that the
butler or the maid or the cook did it that
situation is ruled out by premise one. So
premise one tells you at that situation is
not the actual situation. Premise two
tells you that the maid did not do it. So
any situation in which its true that the
maid did it is also not the actual
situation. So this situation is one in
which its true that the maid did it so
that's not the actual situation according
to premise two. This situation is one in
which its true that the maid did it. So
that's not the actual situation according
to premise two. This situation is one in
which its true that the maid did it. So
that's not the actual situation according
to premise two, and this situation is one
in which it is true that the maid did it.
So that's not the actual situation
according to premise two. Premise three
tells you that the cook didn't do it. So,
that rules out any situation in which it's
true that the cook did it. Well, here's a
situation in which it's true that the cook
did it. So, that situation is ruled out by
premise three. And, here's a situation in
which it's true that the cook did it. So,
that situation is ruled out by premise
three. So, premise one rules out this
situation. Premise two, rules out this,
this, this and this situation. And premise
three, rules out this, this, this and this
situation. Well, whats left? The only
situation left that could be the actual
situation is this one. See cause in this
situation, it's t rue that the butler or
the maid or the cook did it just as
premise one tells us. Its false that the
maid did just as premise two tells us, and
its false that the cook did just as
premise three tells us. But, that's the
situation in which it's true that the
butler did it. So, the conclusion that we
can draw, based on the situations that are
ruled out by premises one, two, and three,
is that the actual situation is this one,
and in that actual situation, it's true
that the butler did it. So, the butler did
it That's why the process of elimination
reasoning that we just considered is
valid. If premise one says, the butler or
the maid or the cook did it. Premise two
says the maid didn't do it, and premise
three says that the cook didn't do it.
Then by process of elimination we can draw
the valid conclusion that the butler did
it and this is why. Let me give you
another example of how you can use the
truth table for disjunction in order to
show whether or not the process of
elimination argument is valid. Suppose we
know that Walter is a professional
football player. Well, that means that he
plays either American football, U.S.
Football, or European football, which
Americans call soccer, or Australian rules
football. But now suppose we find out that
Walter does not play American football.
And you conclude from that, that he must
play European football. So you argue as
follows. Premise 1- Walter plays either
American football or European football or
Australian Rules football, premise 2- he
does not play American football and
therefore you conclude he plays European
football. Well, that argument is invalid
and we can use the truth table for
disjunction to show why it's invalid. Look
at this truth table. Premise one, recall,
is that Walter plays American or European
or Australian rules football. So premise
one rules out the situation in which it's
false that Walter plays American or
European or Australian rules football. And
that's all it rules out. It just rules out
the situation in which it's false that
Walter plays an y of those. Premise two,
Walter doesn't play American Football.
That rules out the situation in which it's
true that Walter plays American football.
So it rules out this situation. And rules
out this situation. And it rules out this
situation. And it rules out this
situation. So, premise one rules out the
situation represented at the bottom.
Premise two rules out the situations
represented by these four columns at the
top. So, can we conclude that Walter plays
European football? No. He might play
European football but he might also play
Australian Rules football. He's looked all
the premise one and premise two together
rule out is these five situations. But
there is no three situations that are
possible. In one of them Walter plays both
European football and Australian rules
football. In another one of them, Walter
plays European football, but not
Australian rules football, and in the
third situation, this left open by
premises one and two, it's false that
Walter plays European football but true
that he plays Australian rules football.
So based on the information that premises
one and two give us, we cannot conclude
that Walter plays European football. He
might play Australian rules football
instead. So the argument that you made is
invalid. In the next lecture, we're going
to consider a truth functional connective
that's different from conjunction and
disjunction in the following way. While
conjunction and disjunction are
connectives that can be used to build
propositions out of two or more other
propositions. Negation, the connective
that we'll talk about next time, is the
connective that is used to build new
propositions out of just one single other
proposition. Negation, in other words, is
a connective that you apply to one
proposition to build a second proposition.
And that's what we'll talk about next
time. Now, there's some exercise's for you
to do. These exercises test your
understanding of the truth table for
disjunction and of how the truth table for
dis-junction can be used to determine
whether a part icular argument that uses
disjunction is a valid argument or not.